2020 Fiscal Year Research-status Report
Theory and applications of Stone-duality for quasi-Polish spaces
| Project/Area Number |
18K11166
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| Research Institution | Kyoto University |
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Principal Investigator |
ディブレクト マシュー 京都大学, 人間・環境学研究科, 准教授 (20623599)
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| Project Period (FY) |
2018-04-01 – 2023-03-31
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| Keywords | quasi-Polish space / topology / locale theory / domain theory |
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| Outline of Annual Research Achievements |
We made additional progress on developing the characterization of quasi-Polish spaces as spaces of ideals of a countable transitive relation. In particular, we have fully verified that there is a simple computable procedure that converts any countable transitive relation encoding a quasi-Polish space into countable transitive relations corresponding to the lower and upper powerspaces of the space. These results were announced at CiE 2020. These results are significant because: (1) The inspiration for the constructions originate in domain theory, and these results demonstrate that many domain theoretical constructs extend to this new characterization, (2) The results demonstrate the computability of several important constructions of quasi-Polish spaces, and (3) The powerspaces have important connections to topological semilattice theory and modal logic, and the characterizations provide new constructive approaches to this area.
We also announced at CiE 2020 a "completion" for computable (countably based) topological spaces, which makes clear connections between computable topology and the characterization of quasi-Polish spaces as spaces of ideals. The "completion" itself has connections to the characterization of quasi-Polish spaces as being dual to countably presented frames, but we anticipate important applications in computable topology because it allows one to define computable topological spaces in a very concise way and within relatively weak logical frameworks.
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| Current Status of Research Progress |
Current Status of Research Progress
2: Research has progressed on the whole more than it was originally planned.
Reason
Because of the coronavirus situation over the past year, there were few chances to present research achievements and to collaborate closely with other researchers on new projects, but over the year we made important advancements in understanding powerspace constructions in terms of spaces of ideals, and also made important advancements in our understanding of quasi-Polish regular frames (still in preparation). Therefore, considering the current global circumstances, I think it is fair to say our research has progressed rather smoothly over the past year.
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| Strategy for Future Research Activity |
In addition to the upper and lower powerspaces, we have also found computable translations corresponding to the valuations powerspace. Valuations are an alternative approach to developing measure theory and probability theory, and during previous research we already verified that the basic theory of valuations behaves well in the context of quasi-Polish spaces. Our characterization in terms of spaces of ideals will allow us to look more at the computabilty aspects of these constructions. We have also looked at translations of continuous (computable) functions corresponding to the functorial aspects of the powerspace constructions, and we plan to present these results in the near future.
We also plan to gather together the results we have found on quasi-Polish regular frames, and hopefully submit a comprehensive paper on the subject later this year or early next year.
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| Causes of Carryover |
Our main use of funds is for international and domestic travel to attend conferences for presenting research and for research collaborations. However, because of the coronavirus pandemic, most conferences were either canceled or held online, and so there were no travel expenses.
There is some hope that with vaccinations we can begin traveling again later this year or early next year, and so we will use the remaining funds for traveling or inviting foreign researchers to Japan when that is possible. At this point, it is still difficult to predict how much travel prices will fluctuate once traveling becomes possible again.
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Research Products
(4 results)
